A006154 -id:A006154 - cp's OEIS frontend

A191277 Expansion of e.g.f. 1/(1 - sinh(x)*cosh(x)).

1, 1, 2, 10, 56, 376, 3152, 30640, 338816, 4226176, 58564352, 892337920, 14834994176, 267186021376, 5182147684352, 107689460377600, 2387077442011136, 56219583797886976, 1401949974947889152, 36902741817196871680, 1022494706646806429696

Offset: 0

Vladimir Kruchinin, May 29 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..427

Cf. A006154, A009229.

CoefficientList[Series[1/(1-Sinh[x]*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k),k,1,n);

my(x='x+O('x^30)); Vec(serlaplace(1/(1 - sinh(x)*cosh(x)))) \\ Michel Marcus, Jun 30 2022

a(n) = Sum_{k=1..n} Sum_{i=0..k} (-1)^i*(k-2*i)^n*binomial(k,i), n>0, a(0)=1.
a(n) ~ n! /(sqrt(5)*(log(2+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, Jun 26 2013
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * binomial(n,2*k+1) * a(n-2*k-1). - Seiichi Manyama, Jun 30 2022

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 19, 153, 1561, 19563, 289339, 4932273, 95258161, 2055639123, 49019157859, 1280056939593, 36329281202761, 1113449691889083, 36651273215389579, 1289577677407798113, 48299079453732363361, 1918528841276621473443, 80559757274836073592499

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

G. C. Greubel, Table of n, a(n) for n = 0..395

Cf. A006154, A235135.

Cf. A001147, A001586, A136630, A235131, A380155.

CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ G. C. Greubel, Apr 05 2017

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A330515 Expansion of e.g.f. Product_{k>=1} 1 / (1 - sinh(x^k)).

Original entry on oeis.org

1, 1, 4, 19, 128, 921, 8912, 87109, 1045200, 13195681, 188639312, 2837096637, 47976425576, 837845855185, 16039578298200, 321739841159317, 6911395312352672, 154749452408120385, 3696709758990757856, 91546190261460505453, 2397650607409036823352

Offset: 0

Ilya Gutkovskiy, Dec 17 2019

nonn

Cf. A006154, A203716, A229263, A270294, A270662, A270663, A270664, A270665, A270666, A330514, A330516, A330517.

nmax = 20; CoefficientList[Series[Product[1/(1 - Sinh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 14, 85, 632, 5559, 56352, 645929, 8252352, 116189291, 1786361216, 29764770941, 534082233856, 10264484355103, 210312181051392, 4575364233983057, 105310034714202112, 2556360647841415379, 65261358332774277120, 1747713179543456515749

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A003704, A003724, A006154, A075729, A331607, A331611.

nmax = 20; CoefficientList[Series[Exp[1/(1 - Sinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A006154[n_] := Sum[Sum[(-1)^j (k - 2 j)^n Binomial[k, j]/2^k, {j, 0, k}], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A006154[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A006154(k) * a(n-k).

a(n) ~ exp(1/(2^(3/2) * log(1 + sqrt(2))) - 3/4 + 2^(3/4) * sqrt(n) / sqrt(log(1 + sqrt(2))) - n) * n^(n - 1/4) / (2^(5/8) * log(1 + sqrt(2))^(n + 1/4)). - Vaclav Kotesovec, Jan 27 2020

A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4k+1) a(n-4*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 732, 5166, 41664, 378001, 3810512, 42253926, 511139904, 6698457481, 94535404992, 1429477706286, 23056267551744, 395120495014561, 7169579673404672, 137321623511274246, 2768602189953629184, 58609968225266985241, 1299827736206335767552, 30137364376923272989806

Offset: 0

Ilya Gutkovskiy, Mar 16 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000670, A006154, A243665, A291975, A306347, A352428, A352430.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k + 1] a[n - 4 k - 1], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(4 k + 1)/(4 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\4, x^(4*k+1)/(4*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

E.g.f.: 1 / (1 - Sum_{k>=0} x^(4*k+1) / (4*k+1)!).

E.g.f.: 1 / (1 - (sin(x) + sinh(x)) / 2).

A200309 Expansion of e.g.f.: 1/(cos(x) - x).

Original entry on oeis.org

1, 1, 3, 12, 65, 440, 3571, 33824, 366113, 4458240, 60321091, 897774592, 14576528801, 256391130112, 4856647308787, 98567413125120, 2133825372539585, 49080991762153472, 1195339768057071619, 30729146849826701312, 831545527540481198465, 23627123985544955559936

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

Radius of convergence |x| < r, where r = cos(r) = 0.739085133215160... (A003957).

The continued fraction converges in the whole complex plane, cut along |z|=infinity.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 65*x^4/4! + 440*x^5/5! +...
where 1/A(x) = 1 - x - x^2/2! + x^4/4! - x^6/6! +...
Special values.
A(Pi/5) = 10/(5*(sqrt(5)+1)/2 - 2*Pi) = 5.534081362740...
A(Pi/6) = 6/(3*sqrt(3) - Pi) = 2.920333635550...
A(Pi/8) = 8/(4*sqrt(2+sqrt(2)) - Pi) = 1.882599403781...
A(Pi/10) = 10/(5*sqrt(10+sqrt(20))/2 - Pi) = 1.5701119741529...
A(Pi/12) = 12/(6*sqrt(2+sqrt(3)) - Pi) = 1.4201994774470...

G. C. Greubel, Table of n, a(n) for n = 0..427
S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian).

Cf. A006154, A003957.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(Cos(x) - x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 10 2018

CoefficientList[Series[1/(Cos[x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

{a(n)=n!*polcoeff(1/(cos(x+x*O(x^n))-x),n)}

x='x+O('x^30); Vec(serlaplace(1/(cos(x) - x))) \\ G. C. Greubel, Jul 10 2018

E.g.f.: E(x)=1/(cos(x) - x) = (1-x^2/((x-1)*G(0) + x^3))/(1-x); G(k)= 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 08 2012
E.g.f.: 1/(G(0) - x) where G(k) =  1 - x^2/((4*k+1)*(4*k+2) - x^2*(4*k+1)*(4*k+2)/(x^2 - 4*(k+1)*(4*k+3)/G(k+1) )); - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ n!/((sin(r)+1)*r^(n+1)), where r = 0.73908513321516... is the root of the equation r = cos(r). - Vaclav Kotesovec, Jun 27 2013

A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6241, 113824, 2460277, 61504384, 1746727201, 55545439744, 1955176596517, 75470959673344, 3169939381277761, 143927870364811264, 7024566555751464757, 366742587098140770304, 20394984041632355113921, 1203587891190987380752384, 75124090160952970927512997

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Cf. A006154, A235134.

Cf. A007559, A007788, A136630, A235132.

CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 3^n / (Gamma(1/3) * 2^(1/6) * n^(2/3) * (log(1+sqrt(2)))^(n+1/3)).

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A385304 Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 16, 117, 1096, 12543, 169576, 2644617, 46735936, 922993083, 20145579136, 481555537917, 12511452674176, 351058439096823, 10579734482269696, 340820224678288017, 11687491783287586816, 425075150516293691763, 16343274366458168160256, 662325275389743380902917

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A006154, A385305.
Cf. A380015, A385306, A385308, A385310.
Cf. A001147, A136630, A235134, A364822.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a136630(n, k));

a(n) = Sum_{k=0..n} A001147(k) * A136630(n,k).

a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * log((1 + sqrt(5))/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025

A385305 Expansion of e.g.f. 1/(1 - 3 * sinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 29, 296, 3921, 63904, 1236509, 27700096, 705098241, 20100847104, 634406699389, 21959759364096, 827184049670161, 33684401687855104, 1474548883501060669, 69051807696652599296, 3444499079760040247681, 182339939994632235515904, 10209271857672376613472349

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A006154, A385304.
Cf. A380017, A385307, A385309, A385311.
Cf. A007559, A136630, A235135.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a136630(n, k));

a(n) = Sum_{k=0..n} A007559(k) * A136630(n,k).

a(n) ~ sqrt(Pi) * 2^(1/3) * n^(n - 1/6) / (5^(1/6) * Gamma(1/3) * exp(n) * log((1 + sqrt(10))/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025

A201628 E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).

Original entry on oeis.org

1, 1, 4, 31, 360, 5601, 109568, 2586151, 71555200, 2271961825, 81441188352, 3253620672303, 143361363439616, 6907049546879041, 361245668908466176, 20383791705206338807, 1234336634416972726272, 79843983527411321710401, 5494767253686351671459840, 400863405346004202504321343

Offset: 0

Paul D. Hanna, Dec 03 2011

nonn

The function 1/(1-sinh(x)) is the e.g.f. of A006154, where A006154(n) is the number of labeled ordered partitions of an n-set into odd parts.

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].

Cf. A214223, A201627, A201595, A006154.

{a(n)=n!*polcoeff(1/x*serreverse(x*(1-sinh(x+x^2*O(x^n)))),n)}

{a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}

E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).
a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).
a(n) = A214223(n+1)/(n+1).

cp's OEIS Frontend

A191277 Expansion of e.g.f. 1/(1 - sinh(x)*cosh(x)).

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A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

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A330515 Expansion of e.g.f. Product_{k>=1} 1 / (1 - sinh(x^k)).

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A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

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A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4*k+1) * a(n-4*k-1).

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A200309 Expansion of e.g.f.: 1/(cos(x) - x).

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A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

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A385304 Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).

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A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4k+1) a(n-4*k-1).